Transfer Path Analysis (TPA), also referred to as Noise Path Analysis (NPA), is an experimental technique for identifying the vibro-acoustic transfer paths in a system, from the active system component(s), generating the structural and acoustic loads, through the physical connections and along airborne pathways, to the target point(s) at the passive system component(s) responding to these loads. The acoustic and vibration responses at the target point(s) are expressed as a sum of path contributions, each associated with an individual path and load. For example, for a pressure response p(ω) expressed in the frequency domain, this can be formulated as follows:
                              p          ⁡                      (            ω            )                          =                              ∑                          i              =              1                        n                    ⁢                                    p              i                        ⁡                          (              ω              )                                                          [        1        ]            with pi(ω) the partial pressure contribution of path i, ω the frequency and n the number of paths. A similar equation can be written for the time domain.
The oldest approach to this problem was to use coherence analysis to identify the various contributions, with all problems related to separating partially correlated sources. In the late eighties, an alternative formulation making use of a source-system-receiver model was developed, expressing each of the partial response contributions as the result of an individual structural or acoustic load acting at a localized interface, and a system response to this interface load. This effectively corresponds to cutting the global system at the interface into an active part generating the interface load and a passive part reacting to the interface load. An example thereof is shown in FIG. 1. For structural loads, this cut typically corresponds to physical connection points (e.g. mounts, bushings, subsystem connections, etc.). For acoustic loads from vibrating surfaces or pulsations from nozzles or apertures, a discretisation by omni-directional volume acceleration point sources is usually applied.
This system approach allows making explicit each of the partial contributions as the result of a load acting at each contribution location and a system response function (SRF) between the load location and the considered target response. In case of a pressure response (e.g. interior noise, pass-by noise, etc.), the key formula for the pressure response p(ω) becomes
                              p          ⁡                      (            ω            )                          =                                            ∑                              i                =                1                            n                        ⁢                          S              ⁢                                                          ⁢              R              ⁢                                                          ⁢                                                F                  i                                ⁡                                  (                  ω                  )                                            *                                                F                  i                                ⁡                                  (                  ω                  )                                                              +                                    ∑                              j                =                1                            p                        ⁢                          S              ⁢                                                          ⁢              R              ⁢                                                          ⁢                                                F                  j                                ⁡                                  (                  ω                  )                                            *                                                Q                  j                                ⁡                                  (                  ω                  )                                                                                        [        2        ]            with Fi(ω) (i=1, . . . , n) the structural loads or forces, Qj(ω) (j=1, . . . , p) the acoustic loads, typically volume accelerations, and SRFi(ω) and SRFj(ω) the system response functions between the input loads and target. A similar equation can be written for vibration responses (e.g. seat vibration, steering wheel vibration, etc.). Concise visualizations of the transfer path contribution results allow quickly assessing critical paths and frequency regions, an illustration thereof shown in FIG. 4, and the separation into loads and noise transfer functions is the key to identify dominant causes and to propose solutions (e.g. act on specific load inputs, act on mount stiffness, act on specific system transfer, etc.).
The test procedure to build a conventional TPA model typically requires two basic steps: (i) identification of the operational loads during in-operation tests (e.g. run-up, run-down, etc.) on the road or on a chassis dyno; and (ii) estimation of the SRF's from excitation tests (e.g. hammer impact tests, shaker excitation test, etc.). The procedure is similar for both structural and acoustical loading cases, but the practical implementation is of course governed by the nature of the signals and the loads.
The estimation of the SRF's between input loads and target response(s) is probably the easiest to control well. The typical procedure is to dismount the active system before measuring the SRF's. In case of soft mount connections, one can also obtain good SRF measurements without disassembling the system. The SRF's can be measured either in a direct or reciprocal way. The use of reciprocal measurements (exciting at the target location(s), measuring the response at the interfaces) has two main advantages: (i) only one excitation is needed per target location while the direct approach requires one excitation per input load; (ii) the limited space at the path inputs can lead to direction errors in direct SRF measurements of up to 10 dB, which can be avoided when using reciprocal measurements.
The identification of the operational loads is the main accuracy factor. For the structural excitation case, there currently exist three ways to identify the forces.
The first approach is to measure the forces directly by using dedicated measuring devices such as load cells. But such direct measurement is up to now not possible in the majority of cases as the load cells require space and well-defined support surfaces, which often makes application impractical or even impossible without distorting the natural mounting situation.
In case that the active and passive structures are connected through soft mounts, the so-called mount stiffness method can be used. This method combines the differential operational responses across the mounts and the mount stiffness profiles to estimate the transmitted mount forces. For a mount i, this can be expressed mathematically as follows:
                                          F            i                    ⁡                      (            ω            )                          =                                            K              i                        ⁡                          (              ω              )                                *                                    (                                                                    a                    ai                                    ⁡                                      (                    ω                    )                                                  -                                                      a                    pi                                    ⁡                                      (                    ω                    )                                                              )                                      -                              ω                2                                                                        [        3        ]            with Fi(ω) the mount force, Ki(ω) the mount stiffness profile and aai(ω) and api(ω) the active and passive side mount accelerations. The mount stiffness method is a fast method, but its disadvantage is that accurate mount stiffness data are seldom available and furthermore depend on the load conditions and excitation amplitudes.
The third approach is the inverse force identification method which identifies the operational loads Fi(ω) (i=1, . . . , n) from closeby acceleration indicator responses aj(ω) (j=1, . . . , v) at the passive system side, by multiplying these with the pseudo-inverse of the measured force-acceleration SRF matrix between all force inputs and indicator responses. Mathematically, this is as follows:
                              [                                                                                          F                    1                                    ⁡                                      (                    ω                    )                                                                                                                                            F                    2                                    ⁡                                      (                    ω                    )                                                                                                      ⋮                                                                                                          F                    n                                    ⁡                                      (                    ω                    )                                                                                ]                =                                            [                                                                                          S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                          11                                                ⁡                                                  (                          ω                          )                                                                                                                                                S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                          21                                                ⁡                                                  (                          ω                          )                                                                                                                          …                                                                              S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                                                      n                            ⁢                                                                                                                  ⁢                            1                                                                          ⁡                                                  (                          ω                          )                                                                                                                                                                                S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                          12                                                ⁡                                                  (                          ω                          )                                                                                                                                                S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                          22                                                ⁡                                                  (                          ω                          )                                                                                                                          …                                                                              S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                                                      n                            ⁢                                                                                                                  ⁢                            2                                                                          ⁡                                                  (                          ω                          )                                                                                                                                                          ⋮                                                        ⋮                                                        ⋮                                                        ⋮                                                                                                              S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                                                      1                            ⁢                            v                                                                          ⁡                                                  (                          ω                          )                                                                                                                                                S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                                                      2                            ⁢                            v                                                                          ⁡                                                  (                          ω                          )                                                                                                                          …                                                                              S                      ⁢                                                                                          ⁢                      R                      ⁢                                                                                          ⁢                                                                        F                          nv                                                ⁡                                                  (                          ω                          )                                                                                                                                ]                                      -              1                                ·                      [                                                                                                      a                      1                                        ⁡                                          (                      ω                      )                                                                                                                                                              a                      2                                        ⁡                                          (                      ω                      )                                                                                                                    ⋮                                                                                                                        a                      v                                        ⁡                                          (                      ω                      )                                                                                            ]                                              [        4        ]            
The matrix inversion is done frequency per frequency, e.g. as applied in EP-A-1855270. The number of indicator responses (v) must significantly exceed the number of forces (n), with a factor 2 as a rule of thumb, to minimize ill-conditioning problems when calculating the pseudo-inverse. Such approach of over-determination is well-described in literature. A serious drawback of this method is the need to perform a large number of SRF measurements to build the full matrix. The latter costs a lot of time and is a main bottleneck for industry.
Another approach is applied in U.S. Pat. No. 5,360,080. Here, optimum load ratios and phase differences between a basic engine mount and the other engine mounts are obtained and tabulated on the basis of empirically obtained transfer functions such as to minimize the vibration level at the vibration evaluation points. These optimum load ratios and phase differences are used as a basis for controlling the vehicle vibrations by adjustment of the mount stiffnesses.
Closely related to this approach is the method disclosed in patent JP-A-06033981 to optimize a vibration reducing by an actuator using the transfer function of a digital filter according to a signal related to the transmitted state of vibration.
Another approach for identifying noise transfer paths is to apply active load cancellation at one of the path inputs and measure the change in noise or vibration at the target(s). This approach, disclosed in patent EP0825358A1, is not so widely used.
Today, the main driver for innovations in TPA is the industry's demand for simpler and faster methods. Existing techniques like inverse load identification are very time-consuming. Several attempts have been made to speed up the TPA process. One of the striking examples is the recently developed Operational Path Analysis (OPA) approach. This approach attracts quite some attention as it requires only operational data measured at the path references (e.g. passive-side mount accelerations, pressures closeby vibrating surfaces, nozzles and apertures, etc.) and target point(s). No SRF's need to be measured. Essentially, it is a transmissibility method as known from structural dynamics, characterizing the co-existence relationship between the target response(s) and path references. This method is indeed very time-efficient, but has several limitations. One of the main limitations is the cross-coupling between the path references. Because of the modal behavior of the structure (resonances), a single force in one of the connection mounts causes vibrations at all path references. This cross-coupling effect easily leads to a false identification of significant paths and wrong engineering decisions. Next to this, the OPA method suffers from ill-conditioning problems related to estimating transmissibilities from operational data. These problems lead to unreliable transmissibility estimates in many cases (e.g. coherent inputs, limited number of orders in the data, etc.).
In view of the above limitations, there is still a need for good methods for obtaining vibrational and/or acoustic data.